Problem: What is the average rate of change of $f(x)=\sqrt{x^3-4}$ over the interval $2\le x \le5$ ?
This is the formula for the average rate of change of a function $f$ over the interval $[a,b]$ : $\dfrac{f(b)-f(a)}{b-a}$ We will need to know the values of $f(2)$ and $f(5)$ to find the slope. $\begin{aligned} f(2)&=\sqrt{2^3-4} \\\\ &=2 \\\\\\ f(5)&=\sqrt{5^3-4} \\\\ &=11 \\\\\\ \dfrac{f(5)-f(2)}{5-2}&=\dfrac{11-2}{3} \\\\ &=3 \end{aligned}$ The average rate of change of $f$ over the interval $2\le x \le5$ is $3$. Notice that the average rate of change is calculated just like the slope of the secant line that intersects the graph of the function at the interval's endpoints. ${1}$ ${2}$ ${3}$ ${4}$ ${5}$ ${1}$ ${2}$ ${3}$ ${4}$ ${5}$ ${6}$ ${7}$ ${8}$ ${9}$ ${10}$ ${11}$ ${12}$ $y$ $x$ $(2,f(2))$ $(5,f(5))$ secant line